Unique factorisation of additive induced-hereditary properties

An additive hereditary graph property is a set of graphs, closed under isomorphism and under taking subgraphs and disjoint unions. Let ${\cal P}_1,

…, {\cal P}_n$beadditivehereditarygraphproperties\mathrm{.}Agraph$G$ has property $({\mathcal{P}}_1\circ \mathrm{.}\mathrm{.}\mathrm{.}\circ {\mathcal{P}}_n)$ if there is a partition $(V_1,\mathrm{.}\mathrm{.}\mathrm{.},V_n)$ of $V(G)$ into $n$ sets such that, for all $i$, the induced subgraph $G[V_i]$ is in ${\mathcal{P}}_i$. A property $\mathcal{P}$ is reducible if there are properties $\mathcal{Q}$, $\mathcal{R}$ such that ${\cal P} = {\cal Q} \circ {\cal R}$; otherwise it is irreducible. Mih'{o}k, Semani\v{s}in and Vasky [J. Graph Theory {\bf 33} (2000), 44–53] gave a factorisation for any additive hereditary property $\mathcal{P}$ into a given number $dc(\mathcal{P})$ of irreducible additive hereditary factors. Mih'{o}k [Discuss. Math. Graph Theory {\bf 20} (2000), 143–153] gave a similar factorisation for properties that are additive and induced-hereditary (closed under taking induced-subgraphs and disjoint unions). Their results left open the possiblity of different factorisations, maybe even with a different number of factors; we prove here that the given factorisations are, in fact, unique.